The Basic Rules of Counting

# the r th size nr and n n1 nr r 2 same

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Unformatted text preview: 2 !...nr ! = n n1 , n2 , . . . , nr Same as number of anagrams of n-letter word with n1 repeats of ﬁrst letter, n2 of second, etc.! Example: How many ways to break the class (of 33) into 11 groups of 3? Math 30530 (Fall 2012) Counting September 13, 2013 8 / 12 Splitting a set up into classes of given sizes In how many ways can we split (partition) a set of size n into r parts, with the ﬁrst part having size n1 , the second size n2 , . . ., the r th size nr (and n = n1 + . . . + nr )? r = 2: same as choosing subset of size n1 (what’s left forms n second part of size n2 ), so n1 General r : Two expressions n n1 n−n1 n2 ... n−n1 −...−nr −1 nr n! n1 !n2 !...nr ! = n n1 , n2 , . . . , nr Same as number of anagrams of n-letter word with n1 repeats of ﬁrst letter, n2 of second, etc.! Example: How many ways to break the class (of 33) into 11 groups of 3? 33 33! /11! = ≈ 5.99 × 1020 3, 3, . . . , 3 11!3!11 Math 30530 (Fall 2012) Counting September 13, 2013 8 / 12 Selecting k items from n, WITH REPLACEMENT In how many ways can we pull out k items from among n different, distinct objects, if each time we pull out an item, we note it and put it back? Math 30530 (Fall 2012) Counting September 13, 2013 9 / 12 Selecting k items from n, WITH REPLACEMENT In how many ways can we pull out k items from among n different, distinct objects, if each time we pull out an item, we note it and put it back? Order matters: (we produce a list (ﬁrst item, second, . . ., last)) nk Math 30530 (Fall 2012) Counting September 13, 2013 9 / 12 Selecting k items from n, WITH REPLACEMENT In how many ways can we pull out k items from among n different, distinct objects, if each time we pull out an item, we note it and put it back? Order matters: (we produce a list (ﬁrst item, second, . . ., last)) nk Example: 8 digit numbers with prime digits? 48 Math 30530 (Fall 2012) Counting September 13, 2013 9 / 12 Selecting k items from n, WITH REPLACEMENT In how many ways can we pull out k items from among n different, distinct objects, if each time we pull out an item, we note it and put it back? Order matters: (we produce a list (ﬁrst item, second, . . ., last)) nk Example:...
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## This note was uploaded on 01/31/2014 for the course MATH 30530 taught by Professor Hind,r during the Fall '08 term at Notre Dame.

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